| L(s) = 1 | − 0.531·3-s + 5-s − 0.0955·7-s − 2.71·9-s − 0.337·11-s − 4.49·13-s − 0.531·15-s − 1.68·17-s − 19-s + 0.0508·21-s − 2.51·23-s + 25-s + 3.04·27-s + 6.61·29-s − 6.79·31-s + 0.179·33-s − 0.0955·35-s − 6.28·37-s + 2.38·39-s + 1.06·41-s − 0.325·43-s − 2.71·45-s + 8.94·47-s − 6.99·49-s + 0.895·51-s + 14.4·53-s − 0.337·55-s + ⋯ |
| L(s) = 1 | − 0.307·3-s + 0.447·5-s − 0.0361·7-s − 0.905·9-s − 0.101·11-s − 1.24·13-s − 0.137·15-s − 0.408·17-s − 0.229·19-s + 0.0110·21-s − 0.524·23-s + 0.200·25-s + 0.585·27-s + 1.22·29-s − 1.21·31-s + 0.0312·33-s − 0.0161·35-s − 1.03·37-s + 0.382·39-s + 0.166·41-s − 0.0497·43-s − 0.405·45-s + 1.30·47-s − 0.998·49-s + 0.125·51-s + 1.98·53-s − 0.0455·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175051142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175051142\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.531T + 3T^{2} \) |
| 7 | \( 1 + 0.0955T + 7T^{2} \) |
| 11 | \( 1 + 0.337T + 11T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 6.79T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 + 0.325T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 1.20T + 59T^{2} \) |
| 61 | \( 1 - 1.21T + 61T^{2} \) |
| 67 | \( 1 - 0.111T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 2.47T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.144622311415895354521875402623, −7.22863596899416318616651329834, −6.67895282087895231665386053770, −5.83459042697794440423264121225, −5.28797464049372073219475034098, −4.61290459196152708463568108107, −3.60283436426330752055759599949, −2.61551997333010992654600081429, −2.03512782353930172438065076411, −0.54791363524363720304891799546,
0.54791363524363720304891799546, 2.03512782353930172438065076411, 2.61551997333010992654600081429, 3.60283436426330752055759599949, 4.61290459196152708463568108107, 5.28797464049372073219475034098, 5.83459042697794440423264121225, 6.67895282087895231665386053770, 7.22863596899416318616651329834, 8.144622311415895354521875402623
